synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The Cheeger-Simons classes are complexified secondary invariants.
Under identifying the fundamental class of a hyperbolic 3-manifold $X$ as an element in the Bloch group, the corresponding degree-3 Cheegers-Simons invariant is the complex volume of the 3-manifold, namely the linear combination
of its Chern-Simons invariant and its volume (e.g. Neumann 11, section 2.3, Garoufalidis-Thurston-Zickert 11).
This appears as the action in analytically continued Chern-Simons theory.
(It is, incidentally, also the contribution of a corresponding membrane instanton wrapping a hyperbolic 3-cycle.)
The volume conjecture for the Reshetikhin-Turaev construction states that in the classical limit it converges to the complex volume (MMOTY 02, Conjedtcure 1.2, see also Chen-Yang 15)
Jeff Cheeger, Jim Simons, Differential characters and geometric invariants, in Geometry and Topology, Proceedings of the Special Year, University of Maryland 1983-84, eds. J. Alexander and J. Harer, Lecture Notes in Math. 1167, Springer-Verlag, Berlin, Heidelberg, New York, 1985, pp. 50–80.
Johan Dupont, Richard Hain, Steven Zucker?, Regulators and characteristic classes of flat bundles (arXiv:alg-geom/9202023)
Walter Neumann, Extended Bloch group and the Cheeger-Chern-Simons class, Geom. Topol. 8 (2004) 413-474 (arXiv:math/0307092)
Walter Neumann, Realizing arithmetic invariants of hyperbolic 3-manifolds, Contemporary Math 541, Amer. Math. Soc. 2011, 233–246 (arXiv:1108.0062)
Stavros Garoufalidis, Dylan Thurston, Christian Zickert, The complex volume of $SL(n,\mathbb{C})$-representations of 3-manifolds (arXiv:1111.2828, Euclid)
The volume conjecture (Kashaev’s conjecture) for complex volume is due to
see also
Relation to analytic torsion is discussed in
Varghese Mathai, section 6 of $L^2$-analytic torsion, Journal of Functional Analysis Volume 107, Issue 2, 1 August 1992, Pages 369–386
John Lott, Heat kernels on covering spaces and topological invariants, J. Diff. Geom. 35 no 2 (1992) (pdf)
Last revised on June 4, 2020 at 13:56:52. See the history of this page for a list of all contributions to it.